How the complex quaternions give each of the Lorentz reps of the standard model (Video 7/14)

Transcript

so we just showed that the complex quaternion split into two pieces left and right-handed vial Spinners so the left-handed spinner transforms like this and the right-handed spinner transforms like this so one thing to notice here is that we're not talking about a matrix acting on a column Vector Instead This is always just an algebra acting on itself so this is just the complex quion acting on the complex quion so from here it's really easy to write down direct Spinners is just going to be the sum of these two things and a myON spinner is also easy to Define it's given by a dra spinner plus or minus its complex conjugate but it turns out that we can model more than just Spinners using the complex quaternion we can also model four vectors so this time the complex quaternion split in a different way and we get two four vectors so a contravariant four Vector transforms like this and its covariant counterpart is simply given by the complex conjugate of this so this is something that might actually look familiar to you we often write down four vectors in terms of 2x two matrices now finally we can do this one last time we can split the complex quaternion into two pieces and this time we're going to get a spinner A scaler and a field string tensor so the scaler transforms like this but the scaler happens to be just a complex number so it commutes with everything and it comes out now this L and L Tilda happen to be in inverses of each other so they cancel and this is in fact a scalar now finally the field strength tensor transforms like this so just to summarize using just the algebra of the complex quaternion we're able to describe left and right-handed vial Spinners dra Spinners myana Spinners contravariant and covariant four vectors scalers and the field string tensor put all together what is this this is all of the Lorent representations of the standard [Applause] model